ProcrustesAbstand

ProcrustesAbstand, commonly called Procrustes distance, is a measure of dissimilarity between two configurations of corresponding landmarks after optimal geometric superimposition. It is widely used in shape analysis to compare shapes while removing effects of location, orientation and, in some variants, size.

Formally, let X and Y be two matrices in R^{n×k}, representing n-dimensional coordinates of k corresponding points. After centering each configuration by subtracting its centroid, the Procrustes distance evaluates how close the configurations are under a similarity transform. There are two main variants:

- Non-scaling variant: d = inf_{R ∈ O(n)} ||X − YR||_F, where R is an orthogonal matrix (rotations and reflections allowed). This distance removes translation and orientation but preserves size.

- General (full) Procrustes variant: d = inf_{s>0, R ∈ O(n)} ||X − s YR||_F, which additionally optimizes a uniform scaling factor s.

The minimizers are obtained from the singular value decomposition of X^T Y = UΣV^T, yielding the optimal rotation R* = VU^T. If scaling is included, the optimal scale is s* = trace(Σ)/||Y||_F^2. The Procrustes distance equals the residual Frobenius norm at these optima. A distance of zero indicates congruent shapes up to rotation, reflection, translation, and (for the full variant) uniform scaling.

Origins and usage: the concept stems from Procrustes, the mythological figure, and is central in Procrustes analysis and generalized Procrustes analysis. It is used to compare anatomical shapes, biomolecular structures, and other geometric configurations, and serves as a basis for downstream analyses such as principal component analysis of shapes, clustering, and classification.